Problem 27 - Thomas Calculus

Question

Exercises \(19-28\) tell how many units and in what directions the graphs of the given equations are to be shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=1 / x \quad \text { Up } 1, \text { right } 1 $$

Step-by-Step Solution

Answer

Shifted equation: \( y = \frac{1}{x-1} + 1 \).

Step 1: Break Down the Instructions

We need to shift the graph of the equation \( y = \frac{1}{x} \) up by 1 unit and to the right by 1 unit. Each of these transformations will affect the equation differently. Our task is to find the new equation after these shifts.

Step 2: Horizontal Shift (Right)

To shift a graph to the right by 1 unit, we replace every "\( x \)" in the equation with "\( x - 1 \)". Thus, the equation \( y = \frac{1}{x} \) becomes \( y = \frac{1}{x-1} \) after this transformation.

Step 3: Vertical Shift (Up)

To shift a graph up by 1 unit, we add 1 to the entire function. After shifting right, we have \( y = \frac{1}{x-1} \). Therefore, the equation becomes \( y = \frac{1}{x-1} + 1 \) after shifting vertically.

Step 4: Write the Final Shifted Equation

The final equation after shifting \( y = \frac{1}{x} \) right by 1 and up by 1 is \( y = \frac{1}{x-1} + 1 \).

Step 5: Sketch the Original and Shifted Graphs

Draw the graph of the original equation \( y = \frac{1}{x} \), a hyperbola with vertical and horizontal asymptotes along the axes. Label this graph. Next, draw the shifted graph using the equation \( y = \frac{1}{x-1} + 1 \), shifted one unit right and up, so the asymptotes are at \( x = 1 \) and \( y = 1 \), respectively. Label this graph as well.

Key Concepts

Horizontal Shift

Graph transformations involve adjusting a graph's position along the coordinate plane without altering its shape. One key transformation is the horizontal shift, which moves the graph left or right. To perform a horizontal shift, you adjust the variable within the function. For example, if you have the function \( y = \frac{1}{x} \), moving it right by 1 unit changes the input as follows:

• Replace \( x \) with \( x - 1 \) to move right.
• Resulting equation becomes \( y = \frac{1}{x-1} \).

This shift changes where the graph crosses the axes but maintains its overall hyperbolic shape.

Vertical Shift

Vertical shifts adjust a graph's position up or down on the coordinate plane. For a vertical shift, you change the function’s output. This transformation adds or subtracts a constant from the entire function. Using \( y = \frac{1}{x-1} \) from our previous horizontal shift:

• Add 1 to shift the graph up by 1 unit.
• The equation becomes \( y = \frac{1}{x-1} + 1 \).

This modification elevates the graph by adjusting all output values, aligning the graph higher without altering its shape.

Asymptotes

Asymptotes are lines that a graph approaches but never actually touches. They are crucial in analyzing rational functions, such as \( y = \frac{1}{x} \). In the original function,

• Vertical asymptote: \( x = 0 \)
• Horizontal asymptote: \( y = 0 \)

After the shifts to \( y = \frac{1}{x-1} + 1 \), these asymptotes change:

• New vertical asymptote: \( x = 1 \)
• New horizontal asymptote: \( y = 1 \)

Understanding where asymptotes lie helps with graph sketching and highlights how transformations affect graph positioning.

Rational Functions

Rational functions are functions that involve a ratio of polynomials. Often seen as \( y = \frac{p(x)}{q(x)} \), they can have unique features like asymptotes and undefined points. The function \( y = \frac{1}{x} \) is a classic example, where the denominator defines the vertical asymptote. Transformations, like shifts, modify these without changing the function's rational nature. Recognizing rational functions’ structure helps in predicting and visualizing how graph transformations, such as horizontal and vertical shifts, impact their visuals and behavior.